In reference [1] we defined the
unexpected total return (UTR) risk premium associated with a commercial
loan. We showed that if the lender chooses to hold the loan but to transfer
the associated unexpected return exposure to a counterparty, then the UTR
risk premium for the loan can be identified as the cost of the required
UTR swap under risk-neutral pricing.

In what follows we first show how
one can impute an allocation of risk capital to a loan transaction from
the UTR risk premium associated with the loan and the lenderís hurdle rate
for excess return on capital. We then relate this imputed risk capital
to that derived from the lender's internal risk capital rules and explore
the properties of imputed risk capital through a series of simple examples.
Finally, we provide two explicit portfolio management strategies that can
be used to support loan origination decisions and loan hold/sell decisions.

Loan Transaction Risk Capital

Our point of departure is the following
definition of the unexpected total return risk premium associated with
a loan transaction:

(1)Here, E[V] denotes the expected net
present value of the loan cash flows with respect to the natural probability
measure for risk rating migration process of the borrower and
denotes the counterpart expected net present value of the loan cash flows
with respect to the risk-neutral measure for the borrower rating migration
process. Since the unexpected total return risk premium is the only sense
in which we use the term "risk premium," we will drop the "UTR" designation
in what follows.

Suppose that the lender has set
a fixed hurdle rate for
the return on capital he requires in excess of the risk-free rate .
Let (for )
denote a sequence of valuation times such that
(not necessarily the time of loan origination) and .
We then set

Like the loan risk premium at time ,
the (transaction) risk capital
allocated to the loan at
depends on the current risk grade of the borrower:

Since rating migration is governed
by a stochastic process, future values of the loan risk premium and of
risk capital cannot be predicted with certainty. We therefore have to make
precise the sense in which applying the risk premium to risk capital yields
the return
in excess of the risk-free rate. The requirement we impose is that if the
excess rate of return for capital is ,
then for any initial state
the expected present value of the aggregate dollar amount spent on risk
capital from time
forward must be equal to the corresponding risk premium .

This leads in a natural way to a
recursive procedure for calculating the risk capital values .
For this, we define:

The risk capital values
are then related to the risk premiums by

(2)

(Note: .)
The are based
on expected cash flow present values and can be efficiently determined
using backward recursion methods.

As we have defined it, the risk
capital allocated to a loan transaction is an imputed quantity determined
in combination by (i) the risk premium that is available to support risk
capital and (ii) the cost per unit time to "rent" risk capital. An immediate
question is what relationship this imputed transaction risk capital bears
to that obtained by applying the lenderís internal risk capital rules.

We make the assumption that the
lenderís par credit spreads (both spot and forward) are calibrated to his
internal risk capital rules as applied to standardized (option-free) term
loans of varying maturity and credit quality. For these standardized loans,
the imputed risk capital and that specified by the lenderís risk capital
rules are the same.

The difficulty arises in applying
risk capital rules to loans with structural complexities, such as the option
for the borrower to prepay, grid pricing, and loan covenants. It is likely
that the lenderís risk capital rules lack the resolution to capture such
fine elements of loan structure. We argue that a reasonable alternative
to applying rules that are insensitive to key elements of loan structure
is to impute risk capital in the manner we have described. The rationale
is twofold. First, the risk premium on which imputed risk capital is based
fully reflects loan structure. Second, the use of risk-neutral pricing
in imputing risk capital is consistent with the absence of internal arbitrage
opportunities within the lenderís loan portfolio. We thus take the point
of view that our method for imputing risk capital to loan transactions
is an appropriate way to expand the "repertoire" of the lenderís risk capital
rules.

To this point we have focused on
the par credit spreads a lender derives from his internal risk capital
rules. These internal par credit spreads are likely to differ, perhaps
substantially, from those observed in the loan market. A question then
is what risk capital rules is the market applying to justify its par credit
spreads. The market, of course, does not directly reveal its risk capital
rules. However, a set of such rules is implicit in the market par credit
spreads and market (excess) return on risk capital. (We note that the long-term
average return on capital for the well-diversified "market portfolio" of
corporate stocks as cited in reference [3] is 8.4% over the risk-free rate.)

To get at these imputed market risk
capital rules, consider a hypothetical lender who holds a well-diversified
loan portfolio approximating the "market portfolio," who sets his par credit
spreads equal to those of the market, and who sets his hurdle rate for
excess return on risk capital equal to the market excess return on capital.
If one applies the methods we have described for imputing risk capital
to the lenderís portfolio, one obtains the imputed market risk capital
rules.

In general, a lender should expect
to allocate more capital to a loan than the imputed market capital (and
pay more for that capital than the market rate). By comparing the amount
of risk capital he is required to allocate to a given loan under his internal
risk capital rules with the amount of risk capital the market would impute
to the same loan, a lender can gain some insight into the penalty he pays
for not achieving the same level of portfolio diversification as the market.

Following [1] we take the view that
the lender maintains two separate internal "accounts": an operating account
to manage loan cash flows and a capital account to support the risk inherent
in those cash flows. The operating account can be viewed as paying the
risk premium to the capital account to enter into an internal unexpected
total return swap as previously described. This swap transfers all of the
risk associated with the loan cash flows to the capital account.

We carry the analysis a step further
by merging the financial "statements" of the two accounts in order to determine
the consolidated return on risk capital for the loan. It is this consolidated
return that we refer to as the risk-adjusted return on capital or RAROC
value (denoted )
for the loan.

The various definitions we have
introduced imply the relationships summarized in Table 1 below.

Table 1: Relationship Between
NPV and RAROC Views of Loan Value

Expected Cash Flow Into Capital Account

Excess Return on Risk Capital

Expected Residual Value in Operating Account

Risk Premium:

Hurdle Rate:

Loan Market Value:

Consolidated Cash Flow Value:

RAROC:

0

The most important of the relationships
in Table 1 is that between the NPV view and the RAROC view of loan value:

(3)

Equation (3) states that a par
loan, i.e., a loan that has zero expected net present value under market
(risk-neutral) valuation, is equivalently a loan that yields the lenderís
hurdle rate as its RAROC value. This parity between the NPV and RAROC views
of loan value is a consequence of our assumption that a par loan must yield
the lenderís hurdle rate as return on risk capital.

Numerical Examples

We now turn to some simple numerical
examples to investigate the effect of loan price and structure on risk
capital and RAROC. We restrict attention to an option-free two-year term
loan with $10,000 principal and annual cash flows. We assume a seven passing
grade credit rating system and par credit spreads, a deterministic value
of 40% for loss in the event of default (LIED), and a constant risk-free
rate of 4.5%. We neglect loan origination/monitoring costs and operating
expense. In addition, we take the lender hurdle rate for excess return
on capital to be 15%.

In Table 2 below we show how risk
capital is affected by the remaining loan term. We consider three variations:
(i) the (fixed) risk capital for a one-year loan, (ii) the first year risk
capital for a two-year loan, and (iii) the expected second year risk capital
for a two-year loan.

Table 2:Effect of Remaining
Loan Term on Transaction Risk Capital

Borrower Risk Rating at Loan
Origination

Aaa

Aa

A

Baa

Ba

B

Caa

One-Year
Loan: Risk Capital

$.68

$2.00

$7.33

$14.67

$66.67

$300.00

$733.33

Two-Year
Loan: Year 1 Risk Capital

$.68

$2.00

$7.35

$14.74

$67.04

$301.89

$741.01

Two-Year
Loan: Expected Year 2 Risk Capital

$.97

$2.67

$8.44

$19.45

$78.91

$285.99

$593.22

The first observation based on Table
2 is that for each borrower risk rating at loan origination, the risk capital
for year 1 of the two-year loan is equal to or marginally greater than
the risk capital for the one-year loan. In the first year, both loans are
exposed to the same risk of loss of principal if default occurs. However,
if a first-year default occurs on a two-year loan, there is the added exposure
to the loss (in present value) of the net loan revenue that would otherwise
have accrued in the second year.

We next observe that in the case
of the two-year loan, there is a significant difference between the first
year risk capital and the expected second year risk capital. For initial
rating grades Aaa through Ba, the expected second year risk
capital is higher. For grades B and Caa, the relationship
is reversed, and the first year risk capital is higher. For grades Aaa
through Ba, the dominant effect is that of possible downward migration
(short of default) in year 1, with a resulting increased exposure in year
2. For grades B and Caa, the dominant effect is that of upward
migration (reducing the exposure in year 2) or default (eliminating any
exposure in year 2).

The final point to be made based
on Table 2 is that variation in risk capital with borrower rating is far
more significant than the variation in risk capital with the remaining
term of the loan. One observes that risk capital increases by a factor
of about 1,000 in going from a Aaa borrower down to a Caaborrower,
reflecting the much higher probability of default.

In Table 3 we examine how the risk
premium, initial risk capital, and RAROC for a loan are affected by pricing.
The borrower rating at loan origination is fixed at Caa (the rating
at which the effect of pricing is most pronounced) and the loan spread
is varied in 10 basis point increments about the par spread of 632 bp.

One observes from Table 3 that the
loan risk premium, initial risk capital, and RAROC are each increasing
functions of the loan spread. In each case, the variation is approximately
linear in the bp
neighborhood of the par spread shown. Both the risk premium and the initial
risk capital are seen to be relatively insensitive to changes in the loan
pricing. A 10 bp change in spread induces a $.25 change in risk premium
and a $1.75 change in risk capital.

By contrast, the dependence of RAROC
is pronounced ó about 1.38% increase in RAROC per 10 bp increase in the
loan spread. The overall implication of Table 3 is that very little of
a loan price increase is lost to the associated increase in the cost of
risk capital. Correspondingly, very little of a price decrease is recovered
by the lender through the associated reduction in the cost of risk capital.

So far, we have focused our attention
entirely on the individual loan transaction. To gain further insight into
the workings of risk capital, it is instructive to consider the implications
of our definition of risk capital at the portfolio level.

For this purpose, we imagine two
banks (A and B), such that bank A maintains a portfolio of exclusively
one-year term loans, while B maintains a portfolio of exclusively two-year
term loans. Both banks follow the policy of originating loans only to borrowers
with a specific rating grade. Borrowers who migrate out of the specific
rating class (including those who default) do not have their loans renewed.
Rather, they are replaced by new borrowers with the required rating grade.

Suppose, for example, that banks
A and B each decide to originate loans only to borrowers with risk rating
Ba.
Under our assumptions, at any point in time, bank A will hold a portfolio
made up exclusively of Ba loans. Bank B, on the other hand, will
hold a mixture, in roughly equal proportions, of two-year loans in their
first year (all Ba rated) and two-year loans in their second year
(mostly Ba rated).

The question we address in Table
4 is how the risk capital requirements of the two bank compare. Results
are shown for each possible origination rating grade.

Table 4: Effect of Loan Term
on Portfolio Risk Capital

Value shown is average risk capital
per $10,000 loan

Borrower Risk Rating at Loan
Origination

Aaa

Aa

A

Baa

Ba

B

Caa

Bank
A Portfolio: One-Year Loans

$.67

$2.00

$7.33

$14.67

$66.67

$300.00

$733.33

Bank
B Portfolio: Two-Year Loans

$.83

$2.34

$7.89

$17.09

$72.93

$294.10

$670.72

Interestingly, the results in Table
4 show that which of the two banks would hold more risk capital depends
on the rating class of borrowers to which they originate loans. If that
rating class is in the range Aaa to Ba, then bank B would
require the greater risk capital. If the two banks decide to go down market
and originate loans only to B or only to Caa borrowers, then
bank A would have to hold the greater risk capital.

In the more realistic case where
both banks originated loans to borrowers of all rating classes but maintained
their strict policies with regard to loan term, which one of the two banks
requires more risk capital would depend on the mix of borrower ratings
in the portfolios of the two banks.

The point to be made from Table
4 is that the decision by bank B to write only two-year loans does not
automatically imply that bank B must hold more risk capital than bank A,
which writes only one-year loans. This behavior is consistent with our
open-end
loan portfolio paradigm, under which the lender renews maturing loans and
originates new loans to maintain an effective portfolio composition equilibrium.
Under the open-end portfolio paradigm, risk capital is driven by instantaneous
risk exposure.

This contrasts sharply with the
closed-end
portfolio paradigm, where the lender is presumed to hold a fixed set of
loans until they mature. If bank A and bank B were operating under the
closed-end paradigm and were seeking to liquidate their respective portfolios,
then it certainly would be the case that bank B would have the greater
cumulative credit risk exposure and would be forced to hold the larger
amount of risk capital.

Risk Capital at the Portfolio
Level

The important question remains whether
the aggregate portfolio risk capital that results from these individual
capital allocations at the transaction level is actually adequate to cover
the lenderís exposure to credit risk.

There are various ways for a lender
to judge the adequacy of his portfolio risk capital. One approach is for
the lender to set a target value for his rating agency credit grade and
then determine whether he holds sufficient risk capital to justify that
credit grade. A specific risk horizon is chosen (one year is typical).
Relative to this risk horizon, let
denote the value of the lenderís loan portfolio,
denote the risk capital the lender allocates to his loan portfolio, and
denote the published probability of default for the target risk rating.

Let
be the probability distribution for .
Then the test for the adequacy of the lenderís portfolio risk capital is

,
(4)i.e., the probability must be at most
that the lenderís portfolio risk capital will be insufficient to cover
any shortfall
in the value of his loan portfolio relative to expectation. We denote the quantile
of by .
Following general practice (see reference [4]), we refer to
as the portfolio value at risk (VAR).

One way to quantify the credit exposure
attributable to an individual transaction is to determine its effect on
VAR. For this we define the following quantities (all relative to the risk
horizon):

Then the following relationship
holds

.
(5)If the relative contribution of any
single transaction to the volatility of the overall portfolio is small,
then to a close approximation

(6)

Suppose that
has been chosen so that the amount of portfolio risk capital
produces exact equality in (4) and just earns the lender his targeted credit
rating grade. Then based on (6), the appropriate amount of risk capital
for the lender to allocate to transaction i is

We previously described how to impute
an amount of risk capital to a transaction consistent with the lenderís
targeted hurdle rate. Equation (7) provides an alternative way to assign
risk capital to a transaction based on its relative contribution to the
portfolio exposure. We refer to the risk capital as given by equation (7)
as the exposurerisk capital allocated to the transaction.

Our analysis here in combination
with that in reference [1] supports two separate tests to apply to a transaction:

value test: originate (or
buy) a transaction if it has a positive expected net present value under
the risk-neutral measure calibrated to the market par credit spreads.

exposure test: hold transaction
in portfolio if its exposure risk capital is less than or equal to its
imputed risk capital.

Note that in applying the value test,
it is the market par credit spreads that are relevant since value is measured
in relation to market pricing. A "good" loan to originate is one that the
lender should be able sell at a profit. In applying the exposure test,
it is the lenderís internal par credit spreads that are relevant. A "good"
deal to hold is one that is "risk efficient" in the sense that the actual
exposure it adds to the lenderís portfolio is at most that implied by the
lenderís internal pricing of risk. Since imputed risk capital and exposure
risk capital are equal for the standardized loans used to calibrate the
lenderís internal par credit spreads, the standardized loans will necessarily
pass the exposure test.

We observe that a transaction can
pass the value test but fail the exposure test. The incentive then is for
the lender to originate (or buy) the transaction but then sell it for a
quick profit. The lender does not want to hold the deal in his portfolio
because he does not expect to be adequately compensated for the incremental
exposure it adds to his portfolio.

It is also the case that certain
transactions in the lenderís portfolio can pass the exposure test but fail
the value test. For such a deal, the expected loss is a sunk cost. The
lender can realize the loss by selling the deal immediately or defer the
loss by holding onto the loan. If the lender chooses to hold the deal,
he will at least be adequately compensated for the incremental risk capital
it ties up.

It is certainly possible that many
(perhaps most) of the transactions in a given lenderís portfolio will fail
the exposure test. That would suggest an inconsistency between the return
on capital required by the lenderís shareholders and the "risk efficiency"
of the portfolio. Given the risk characteristics of the lenderís portfolio,
the lenderís shareholders may be unwilling to accept a lower return on
risk capital than the target hurdle rate. The lender would then be obliged
to eliminate from his portfolio those loans that are the most "risk inefficient"
in an effort to improve the portfolio return on capital.

Possible Extensions

There are a number of directions
in which our approach to measuring credit risk can be extended.

We have assumed that LIED is deterministic.
The calculation of the risk-neutral measure for credit rating migration
then follows reference [6]. However, we have been able to extend the methods
of [6] to the case in which LIED is a random variable with known mean and
variance. The definitions of risk premium, risk capital, and RAROC then
directly carry over.

The risk-free rate used to discount
cash flows has been assumed fixed. It is possible to combine an arbitrage-free
stochastic interest rate model (such as the Heath-Jarrow-Morton model in
reference [7]) with the arbitrage-free credit migration model we have described
to obtain a joint model for interest rate risk and credit risk. Again,
our basic definitions carry over without change. However, with more state
variables required to describe the stochastic evolution of loan value,
the joint model entails a significant increase in computation.

Our focus has been primarily on
commercial loans. However, the framework we have described for measuring
credit risk extends to bonds, credit derivatives, and generally to any
structured set of cash flows subject to credit risk. The key requirement
is that one must be able to calculate the expected net present value of
those cash flows under both the natural and risk-neutral rating migration
process measures in order to obtain the associated risk premium.

The authors are currently working
on a factor model for credit rating migration that builds on the continuous
state model for risk rating described in reference [8]. The model incorporates
a single "business cycle" factor that drives the stochastic evolution of
rating grade transition probabilities, LIED distribution parameters, and
par credit spreads. It also accounts for the correlation between the credit
migration processes of different borrowers without imposing the requirement
that the probability distribution for portfolio value at the risk horizon
be Gaussian.

Summary

We have introduced specific definitions
for three important quantities related to the measurement of credit risk
at the transaction level: risk premium, risk capital, and risk-adjusted
return on capital (RAROC). The key element in our approach is linking (through
risk-neutral pricing methods) the marginal pricing of credit risk implicit
in one-period par credit spreads to the pricing of the aggregate risk inherent
in a multi-period credit instrument.

We have identified the risk premium
associated with a credit transaction as the cost of a form of total return
swap which transfers the unexpected return risk on the trans-action to
a counterparty (which could be the lenderís risk capital account). This
risk premium supports an imputed amount of risk capital consistent
with the lender realizing a targeted hurdle rate as excess return on capital.
We differentiated this imputed transaction risk capital from the exposure
risk capital allocated to the transaction based on consideration of
the risk characteristics of the lenderís credit portfolio and his targeted
credit rating. We defined the RAROC associated with a loan as the excess
return on imputed risk capital generated by the loan revenue. We then explored
the properties and interrelationships of these risk quantities through
a series of simple examples based on a two-year term loan.

Two loan portfolio management guidelines
emerged from our analysis: (1) originate (or buy) loans with positive expected
net present value under the risk-neutral credit migration process measure
(i.e., loans for which )
and (2) hold those loans for which exposure risk capital is less than or
equal to imputed risk capital.